On the existence of graphs which can colour every regular graph

Let $H$ and $G$ be graphs. An $H$-colouring of is a proper edge-colouring $f:E(G)\rightarrow E(H)$ such that for any vertex $u\in V(G)$ there exists $v\in V(H)$ with $f\left (\partial_Gu\right )=\partial_Hv$, where $\partial_Gu$ $\partial_Hv$ respectively denote the sets edges in incident to vertices $u$ $v$. If admits an we say colours $G$. The question whether graph every bridgeless cubic addressed directly by Petersen Colouring Conjecture, which states graph. In 2012, Mkrtchyan showed if this conjecture true, unique connected can colour all paper extend show were remove degree conditions on $H$, coloured substantially only other graph: subcubic multigraph $S_{4}$ four vertices. A few similar results are provided also under weaker assumptions second part paper, consider $H$-colourings regular graphs having strictly greater than $3$ that: (i) $r>3$, does not exist (possibly containing parallel edges) $r$-regular multigraph, (ii) $r>1$, $2r$-regular simple